Download SDR equivalents to analog functions in a small-scale radio

SDR equivalents to analog functions in a small-scale radio telescope
Marcus Leech, Science Radio Laboratories
Introduction
Small-scale radio astronomy has typically been accomplished using simple analog circuitry, often
constructed by the observer themselves from scrap-box parts. Observational goals are typically
quite modest, and those modest goals are often reflected in the relatively-primitive analog receiver
systems used.
With the advent of affordable Software Defined Radio technology comes an opportunity to reexamine the methods and techniques traditionally used by an amateur observer to construct a
suitable radio telescope receiver.
Analog Total Power Receiver
In a typical small-scale observatory conducting total-power observations, the receiver is typically a
single or dual-conversion superheterodyne design, with the final IF stage driving a hardware squarelaw detector, integrator, and DC amplifiers, as shown below:
The signal arrives from the antenna, and is amplified by one or more stages of RF amplification (and
sometimes bandpass filterig), and is then presented to the 1st mixer where the signal is converted to
a fixed intermediate frequency by the mixing action between the incoming signal frequency and the
frequency produced by the Variable Frequency Oscillator. The result is a fixed intermediate
frequency that is filtered, and amplified by the 1st IF amplifier. The signal is then typically mixed
again in the 2nd mixer stage, and converted to the final intermediate frequency with a fixed Local
Oscillator. Once the final IF is produced, it is detected with a square-law detector, usually a
detector diode. The detector produces an output voltage that is proportional to the incident AC
power. This voltage is then amplified by the DC amplifier, and low-pass filtered with an integrator
stage. The output of the integrator is then usually fed to a recording device of some sort.
Digital, SDR-based Total Power Receiver
In a digital receiver, the signal path is typically identical up to the output of the 1st IF amplifier, with
some crucial differences. In an SDR-based receiver, the incoming signal is typically converted to a
quadrature-baseband signal, producing the so-called ‘I’ (in-phase) and ‘Q’ (quadrature) signals.
This is achieved using two mixers that are fed with the RF signal, and a local-oscillator that has
both a sine and cosine output. This is shown below:
Observe that there are two mixers, instead of the usual single mixer. There is also a so-called
phase-splitter that produces both sine and cosine components from a single sine-component LO
signal. It is also the case that the VFO is set so that Flo is equal to Frf, that is, the VFO operates
at the same frequency as the desired reception center frequency. This is a so-called directconversion receiver. A direct-conversion receiver without so-called quadrature conversion suffers
from a near-fatal flaw, in that both the sum and difference components in mixing perfectly overlap.
In some types of radio-science, that would be acceptable, but in order for a direct-conversion
design to “act” like a conventional receiver, a different signal representation is required. The I/Q
signal is mathematically like a complex number with both a real and imaginary component. This
complex (I/Q) representation allows us to disambiguate the sum/difference frequencies when we
process the signal mathematically.
Consider an input signal a F, with a bandwidth of Bw, when that signal is processed by a directconversion quadrature mixer, the resulting baseband signal occupies a new spectrum that spans from
-Bw/2 to +Bw/2. Now in the real world, there’s no such thing as a negative frequency, but the
I/Q, or complex representation allows us to pretend that there is. It’s a mathematical abstraction
that turns out to be incredibly useful in the real world.
Once we have our I and Q signals, they are typically low-pass filtered in hardware (analogous to the
IF filters in a conventional receiver), amplified, and then sampled with a high-speed ADC.
The low-pass filters in a quadrature-digital receiver are also known as anti-alias filters. Recall the
Nyquist Sampling Theorem that states that a signal of bandwidth B may be adequate sampled (and
later reconstructed) using a sample rate of no more than 2B. Put another way, a digital sampler,
such as an ADC, operating at a sample rate of Fs can adequately sample a signal of bandwidth Fs/2.
Any signal beyond Fs/2 becomes aliased into the input pass-band. Hence the need to have an antialias filter in front of the sampler in order to reduce frequencies above Fs/2. Many digital receivers
use a fixed sampling rate, in order to simplify anti-alias filters, and it is usually the case that 5th or
7th order passive filters are used prior to sampling. In some variable-bandwidth designs, the antialias filters are based on a switch-capacitor filter technology that allows adjusting the low-pass
corner frequency. This is how variable-rate sound cards for PCs perform anti-alias filtering.
Once the signal has been sampled at some rate that is convenient, it may be further processed in
digital hardware to reduce the sample rate, via a technique called decimation. In an agile digital
receiver, it is typically the case that the signal is sampled at a high rate that is commensurate with
available hardware budget, and then decimated according to the end application.
Once the signal has been decimated (which is also a form of digital low-pass filtering), it is
presented to the final digital signal-processing chain for further processing and information
extraction.
The digital processing chain
Once the signal has been digitized, and possibly decimated, it is presented to a general-purpose
digital signal processing chain. We’ll assume that chain “lives” inside a general-purpose computer.
We’ll assume that our signal arrives in the I/Q format described above, but for direct-sampled
receivers, the signal arrives as a single quantity. Such direct-sampled receivers are typically used
only when the RF frequencies are relatively low (in the HF bands, typically).
Our signal arrives in the computer as I/Q samples that represent a swath of bandwidth (Bw) from
Fc-(Bw/2) to Fc+(Bw2), where Fc is the center frequency of interest. Keep in mind, though, the
signal is in baseband form, with Fc at 0Hz.
Once we have the signal it can be further filtered, and even decimated further. It is sometimes the
case that the sample rates offered by the front-end hardware don’t precisely match the sample
rates desired in the application, so further decimation is often performed within the computer.
Filters in a digital receiver are implemented typically with FIR or FFT topology filters, which give
fairly-precise responses, with perfect symmetry, and flat passband responses. The design of such
filters (or rather, the coefficients for such filters) is beyond the scope of this document, but tools
exist for designing such filters.
Assuming that our signal is filtered appropriatley, we need to detect it—that is, extract a number
that is proportional to the incident power. In a digital receiver, we simply calculate:
X = (I*I) + (Q*Q)
Recall that the power flowing in a circuit is proportional to the square of the voltage across that
circuit. Our digital signals are numerical samples of an RF voltage, and so detection is
mathematically straightforward. Once the signal has been detected, it needs to be integrated (lowpass filtered). In an analog receiver, the integrator would likely be constructed using an op-amp RC low-pass filter. In a digital receiver, we use the single-pole IIR filter as a low-pass filter:
Y = (a * X) + ((1.0 – a) * Y)
As shown, Y represents the filter output, X the incoming signal from the detector, and ‘a’ is a
parameter that determines the corner low-pass cutoff frequency (or, in more-familiar terms, the
integration time constant).
Once the signal has been detected, and integrated (low-pass filtered), we can reduce its sample rate
to a rate that is twice the low-pass corner frequency of the integrator. In a total-power radiometer,
we typically use integration times from about 1 second up to several 10s of seconds. For
consistency reasons, it’s typical to reduce the post-integrator sample rate to some fixed value, like
2Hz.
Once the signal has been integrated and decimated down to a standard sample rate, we can scale it
and offset it with simple arithmetic, before it is logged to an external file for later analysis and
processing.
For example, if we wanted to convert the signal Y above (output of the integrator) to Jansky, we
would determine an offset value, and a coefficient value to apply to the signal prior to logging it:
Y1 = (Y + offs) * Coeff
Where ‘offs’ and ‘Coeff’ would need to be determined empirically for each situation. It is usual,
however, to apply such scaling factors in a post-processing step, and that is also true for analog
receivers.
An analog spectrometer
Recall the diagram of the analog total-power receiver given earlier. It has a VFO (VariableFrequency Oscillator) that determines the center-frequency of operation, and two or more stages of
Intermediate Frequency (IF) conversion and filtering.
If that receiver is modified so that there’s a very-narrow 3rd IF stage, with tightly-controlled analog
filter and a precision detector, then by varying the VFO frequency in a swept-frequency fashion,
and recording the outputs of the narrow-band 3rd IF detector, we can measure the frequency
spectrum of a signal. Since the 3rd IF detector is very narrow, it allows us to estimate the power
only over a tiny frequency portion of the incoming signal. By synchronizing the detector output with
the so-called sweep generator, we can produce a spectral estimate.
A digital spectrometer: The Fourier Transform
Back in the 19th century a French mathematician and physicist, Joseph Fourier, invented a new
mathematical technique for the analysis of complex vibrations and heat transfer. Today his keen
mathematical insights into vibratory processes has revolutionized the way signals are analyzed and
processed.
Various versions of the so-called Fourier Transform exist today, but for discrete-time-sampled
digital signals, the technique most often applied is the Fast Fourier Transform (FFT).
Computing an FFT over a block of discrete-time-samples yields an estimate of the frequencyspectrum of that signal.
Consider our incoming I/Q signal from our ADC, that occupies a bandwidth of Bw Hz. If we
calculate an FFT of length Z on the signal, we require Z samples of that signal to produce a spectral
estimate over those samples:
O[0..Z] = FFT(S[0]..S[Z])
Where S is our incoming I/Q signal as a complex number. If our incoming signal has a bandwidth of
Bw, then each “bin” of the output spectrum represents Bw/Z Hz of bandwith. If our signal
bandwidth is 1MHz, and our transform length is 1024, then each “bin” in the FFT output
represents 976.25Hz of the incoming signal.
For a spectrally-stationary signal, frequency estimates become better as you integrate the output
bins, and so it is typically the case that each FFT output bin is run through a single-pole IIR filter,
just as in the total-power receiver:
E[i] = ((O[i] * a) + ((1.0 – a)*E[i]), i=0..Z
Where E[x] is the integrated estimate for bin ‘x’, O[x] is the FFT output for bin ‘x’.
Modern FFT implementations on modern desktop-CPU hardware can execute extremely quickly,
which means that the resolution of the spectral estimates can be extremely high. SETI processing,
for example, requires very high resolution, and thus very large FFTs. Such resolution would be
difficult to obtain in an analog spectrometer. It is not unusual to find digital spectrometers in a
Software Defined Radio to have FFTs with many 10s of thousands of FFT bins, giving very fine
spectral resolution, capable of measuring very small doppler shifts, for example.
Analog Pulsar Processing: the Analog Filter Bank
Pulsars are rapidly-rotating neutron stars whose intense magnetic filed causes a beam of
electromagnetic radiation to emanate from each pole. When that beam intersects our antenna, we
receive a brief pulse as the pulsr beam transits the beam of our antenna.
The peak power of these pulses, as received on earth is fairly weak, so they must be observed over
significant bandwidths, in order to achieve adequate sensitivity. The problem is that the interstellar
medium is dispersive in nature, and thus pulses at lower frequencies arrive after pulses at higher
frequencies, which means that the pulses tend to become smeared, when observed over the entire
bandwidth. The smearing is often so severe that the pulses get smeared below the noise floor of the
detection apparatus.
In analog pulsar receivers a so-called filter-bank is applied to the 2nd IF to divide up the incoming
bandwidth into small sub-bands, like so:
The resulting sub-bands are individually detected, and then delayed in a variable analog delay line,
before being summed in an analog summing amplifier. In this way, all the pulses in the various subbands can be made to arrive simultaneously at the summing amplifier, and thus produce a nonsmeared (or dramatically-less-smeared) pulse profile at the output of the summing amplifier.
Digital Pulsar Processing: Coherent De-dispersion
In 1975, Hankins and Rickett1 determined the transfer-function of the interstellar medium. They
were able to predict the so-called dispersion measure along any line of sight, knowing only the
column depth of the medium (the distance to the observed object).
What this meant was that you could construct a digital filter to undo the effects of the interstellar
medium, prior to detection of the signal. They published an algorithm (a series of mathematical
steps) for determining the required coefficients of a complex FFT filter that would de-disperse the
incoming signal, as long as you knew the dispersion measure (DM) along the line of sight, the
observing bandwidth, and the observing center frequency.
1 See: Hankins and Rickett, Pulsar signal Processing, Methods in Computational
Physics Vol 14, 1975.
It turns out that the required filters are quite large, when attempted over significant bandwidths, so
it wasn’t until fairly recently that de-dispersion in real time has been attempted by radio
astronomers, since the necessary computing power is significant.
In a digital pulsar receiver, with real-time de-dispersion, the signal is processed like so:
Y = FFT_FILTER(S, DM, Fc, Bw)
Where Y is the de-dispersed signal, S is the incoming signal, DM is the dispersion measure, Fc is
the center frequency, and Bw is the observing bandwidth. Once you have the de-dispersed signal, it
can be processed with a detector, just as in a total-power receiver, although the output sample
rates are typically much higher, on the order of several Khz, because pulsar pulses are quite
narrow, although their repetition rates are usually quite long relative to their pulse widths.
Using this mathematical model, you can produce pulse profiles that are very accurate renderings of
what the pulse shape actually looks like, without any appreciable smearing in time. Having highfidelity pulse profiles allows astrophysicists to learn more about the nature of pulsars, neutron stars,
and astrophysics in general.
Pulsar Folding: Digital Synchronous Detection
Recall that pulsar pulses are relatively weak, and exceedingly narrow. This means that the total
energy captured by a radio telescope during a pulsar pulse is very small. We’ve already discussed
how we use de-dispersion to reduce or eliminate smearing of a pulse, and thus improve the odds of
detection.
Another technique that is used is so-called folding of the detected pulsar pulses. By knowing fairly
precisely what the pulsar repetition rate is, we can add individual pulses together to produce a
composite pulse profile with a higher SNR than a single pulse. Achieving this digitally is fairly
straightforward, since you can manipulate samples in a fixed-length buffer that represents a single
pulse period, and add these buffers together over time, to produce an integrated pulse profile.
The analog approach to this problem would be to use a series of analog delay lines and a summing
amplifier with an appropriate integrator. But even in analog pulsar receivers, it is typically the case
that the detector(s) output are sampled with a ADC with a sample rate of a few Khz, and so-called
folding is achieved digitally, since the analog approach would be very cumbersome, and ADCs with
Khz sample rates have been available for decades.
Transient Detection
Of some astrophysical interest are so-called cosmic transients—powerful pulses much like pulsar
pulses, but random, sporadic, and usually extremely strong. Sometimes, these pulses are caused by
extraordinary outbursts from existing and known pulsars, and sometimes their origins aren’t clear.
Transient detection requires that you be able to sample the detector output at a fairly high rate,
since transient pulses are quite narrow, similar to pulsar pulses, and that you be able to detect an
event that qualifies as a transient, according to your criterion. The output of a detector in a radio
telescope is very noisy both due to inherent noise in the electronics, and the noise of the cosmos
itself. The task of transient detection lies in determining when a brief noise burst is statistically
significant, and recording the transient and surrounding “context”.
With digital signal processing, applying such statistical tests is relatively straightforward, and well
understood. You have to maintain a running estimate of the standard deviation of the incoming data,
and “trigger” when a detected event exceeds the standard deviation by some configurable
threshold. Quite straightforward, and cheap to compute, even at rates of several 10s of Khz on a
modern computer.
A further refinement in transient detection is to use multiple receiver chains, which originate in
different sky positions, and use the technique of anti-correlation to remove transients that are
clearly of local origin (because they appear in one or more of the receiver outputs at the same time).
Any transient event that is detected in only a single receiver is more likely to be of cosmic origin
than one that is detected in more than one receiver.
It would be unlikely that you would construct a transient detector in a strictly-analog receiver, the
required circuitry would be “finicky” and hard to characterize, but it is certainly theoretically
possible.